All of the 3rd grade teachers and students from Covington went on a field trip to an art museum. Tickets were $$5.50$ each for teachers and $$4.50$ each for students, and the group paid $$58.00$ in total. The next month, the same group visited a science museum where the tickets cost $$16.50$ each for teachers and $$12.00$ each for students, and the group paid $$162.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5.5x+4.5y = 58}$ ${16.5x+12y = 162}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-16.5x-13.5y = -174}$ ${16.5x+12y = 162}$ Add the top and bottom equations together. $ -1.5y = -12 $ $ y = \dfrac{-12}{-1.5}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {5.5x+4.5y = 58}$ to find $x$ ${5.5x + 4.5}{(8)}{= 58}$ $5.5x+36 = 58$ $5.5x = 22$ $x = \dfrac{22}{5.5}$ ${x = 4}$ You can also plug ${y = 8}$ into $ {16.5x+12y = 162}$ and get the same answer for $x$ ${16.5x + 12}{(8)}{= 162}$ ${x = 4}$ There were $4$ teachers and $8$ students on the field trips.